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- Title
On the maximal displacement of subcritical branching random walks.
- Authors
Neuman, Eyal; Zheng, Xinghua
- Abstract
We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each $$n\in \mathbb {N},$$ let $$M_{n}$$ be the rightmost position reached by the branching random walk up to generation n. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists $$\rho >1$$ such that the function satisfies the following properties: there exist $$0<\underline{\delta }\le \overline{\delta } < {\infty }$$ such that if $$c<\underline{\delta }$$ , then while if $$c>\overline{\delta }$$ , then Moreover, if the jump distribution has a finite right range R, then $$\overline{\delta } < R$$ . If furthermore the jump distribution is 'nearly right-continuous', then there exists $$\kappa \in (0,1]$$ such that $$\lim _{n\rightarrow \infty }g(c,n)=\kappa $$ for all $$c<\underline{\delta }$$ . We also show that the tail distribution of $$M:=\sup _{n\ge 0}M_{n}$$ , namely, the rightmost position ever reached by the branching random walk, has a similar exponential decay (without the cutoff at $$\underline{\delta }$$ ). Finally, by duality, these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior.
- Subjects
RANDOM walks; MAXIMA &; minima; PROBABILITY theory; DISTRIBUTION (Probability theory); EXPONENTIAL decay law
- Publication
Probability Theory & Related Fields, 2017, Vol 167, Issue 3/4, p1137
- ISSN
0178-8051
- Publication type
Article
- DOI
10.1007/s00440-016-0702-8